= Classification of subgroups of a dihedral group
{title2=$\langle r^d\rangle,\quad\langle r^d,r^js\rangle\ (d\mid n)$}
Write $D_{2n}=\langle r,s:r^n=s^2=1,\ srs=r^{-1}\rangle$. A subgroup contained in the rotations is a <cyclic subgroup> $\langle r^d\rangle$, with $d\mid n$. If it contains a reflection, every other reflection differs from a chosen $r^js$ by a member of the rotational intersection, so it is $\langle r^d,r^js\rangle$, with $j$ specified modulo $d$. These subgroups have orders $n/d$ and $2n/d$ respectively. This gives a complete list, including the trivial and whole subgroups; see https://maths.qmul.ac.uk/~pjc/MTH714U/so3.pdf[Peter Cameron's group-theory exercise solutions, question1].
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